Abstract/Summary

The Gauss–Newton algorithm is an iterative method regularly used for solving
nonlinear least squares problems. It is particularly well suited to the treatment of very large scale
variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure
consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is
solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants,
the algorithm is attractive because it does not require the evaluation of second-order derivatives in
the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive
to apply operationally in meteorological forecasting, and various approximations are made in order to
reduce computational costs and to solve the problems in real time. Here we investigate the effects on
the convergence of the Gauss–Newton method of two types of approximation used commonly in data
assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least
squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods
where the true linearized inner problem is approximated by a simplified, or perturbed, linear least
squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton
methods converge and also derive rates of convergence for the iterations. The results are illustrated
by a simple numerical example. A practical application to the problem of data assimilation in a
typical meteorological system is presented.